On the real angular stability radius for sectorial matrices
Abstract
The real stability radius problem asks for the smallest real perturbation that makes a given matrix lose rank, and it is a structured counterpart of the corresponding complex stability radius problem. In this paper, we consider an angular version of this question for sectorial matrices, where the size of a perturbation is measured by its largest phase instead of its largest singular value. More precisely, for a sectorial matrix M, we investigate the minimum required largest phase of a real and sectorial perturbation Δ for which I+MΔ loses rank. We show that for some matrices, the complex and the real angular stability radius are the same, but that they are in general different, such as when M is diagonal or complex symmetric. For these cases, we derive a formula for the real angular stability radius, and this formula depends on the two largest phases of M. Moreover, in these cases we show that the result for n × n matrices can be reduced to that for 2 × 2 matrices by compressing an arbitrary real destabilizing perturbation to the real two-dimensional subspace spanned by the real and imaginary parts of a destabilizing vector.
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BibTeX
@misc{khong2026real,
title = {{On the real angular stability radius for sectorial matrices}},
author = {Sei Zhen Khong and Xin Mao and Li Qiu and Axel Ringh},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30843},
eprint = {2608.30843},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30843},
}
RIS
TY - UNPB
TI - On the real angular stability radius for sectorial matrices
AU - Sei Zhen Khong
AU - Xin Mao
AU - Li Qiu
AU - Axel Ringh
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30843
UR - https://arxiv.org/abs/2608.30843
AB - The real stability radius problem asks for the smallest real perturbation that makes a given matrix lose rank, and it is a structured counterpart of the corresponding complex stability radius problem. In this paper, we consider an angular version of this question for sectorial matrices, where the size of a perturbation is measured by its largest phase instead of its largest singular value. More precisely, for a sectorial matrix M, we investigate the minimum required largest phase of a real and sectorial perturbation Δ for which I+MΔ loses rank. We show that for some matrices, the complex and the real angular stability radius are the same, but that they are in general different, such as when M is diagonal or complex symmetric. For these cases, we derive a formula for the real angular stability radius, and this formula depends on the two largest phases of M. Moreover, in these cases we show that the result for n × n matrices can be reduced to that for 2 × 2 matrices by compressing an arbitrary real destabilizing perturbation to the real two-dimensional subspace spanned by the real and imaginary parts of a destabilizing vector.
ER -